A wedding is very special, and although a picnic/BBQ sounds wonderful for the occasion, to just invite guests to bring a dish seems a bit informal. Perhaps you should identify select guests to bring the food and suggest different dishes.
My family and some of my husband's relatives catered my wedding reception, of course that was pretty much 20 years ago. It turned out beautiful...there were about 10-12 turkeys, 35-40 boxes/pans of chicken, several homemade BBQ dishes, an assortment of green vegetables (cabbage, green beans, mixed greens, and salads (including 8 different pasta dishes), and all kinds of deserts (cakes, pies). The food was absolutely delicious, and those who did not cook brought beverages. For a moment there, I almost thought everyone in attendance brought a dish. It was definitely a memorable occasion. But I created a menu and personally went to each individual and asked if they would make a dish for our wedding. By handling it this way, even persons I had not approached brought something. The word-of-mouth effect obviously came into play. Nevertheless, we had enough food to feed a whole city, as opposed to the 250 guests invited.
Perhaps you too should start by putting together a menu for the occasion, and based on it, identify 2-3 persons to be responsible for each item. But make certain you approach and make a personal request of the individuals. For example and depending upon the number of guests you anticipate, you might identify and ask: 3-4 people to bring chicken; 3-4 on burgers; 3-4 for hotdogs; 3-4 for veggies, i.e. greens, salads, corn-on-the-cob; 2-3 on bread, etc. The number of guests you anticipate, should determine how many persons you ask to be responsible for what items.
If handled in the manner above, you do not have to hold off on distributing your invitations. Please know, for all practical purposes, this type of request is best made in person.
R. (Prince George, VA)